Estimate π's Monte Carlo Variance
Monte Carlo variance for estimating pi is a medium quant interview question on Expected Value, reported to have been seen at Goldman Sachs.
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This Monte Carlo question is about estimating a fundamental constant from random sampling in a geometric setting, then quantifying how noisy that estimate is. It connects simulation, expected value, and variance, using uniform samples and an indicator construction that every serious quant candidate should be comfortable with. Although the surface is geometry, the core is probability and statistics.
It trains your understanding of Bernoulli random variables, sample means, and how variance behaves under averaging and scaling. It reinforces skills in expressing Monte Carlo estimators formally, computing their variance, and interpreting how the error decreases with more samples. It also develops intuition about probabilistic error bounds in simulation-based quant prep.
This matters for quant interviews because real pricing, risk, and XVA engines rely heavily on Monte Carlo. Interviewers want evidence you can analyze estimator quality, not just implement simulations.
What it tests
When estimating a probability or proportion using independent Bernoulli trials, the variance of the sample mean estimator is governed by the binomial variance formula: for each trial, the variance is $p(1-p)$, and for $N$ trials, the variance of the mean is $p(1-p)/N$. If the estimator is a scaled version of this mean (as when estimating an area or a probability times a constant), the variance is scaled by the square of that constant. This structure arises because the variance of a sum of independent random variables is the sum of their variances, and scaling a random variable by a constant multiplies its variance by the square of that constant. This principle holds for any Monte Carlo estimator based on counting the fraction of successes in independent trials, regardless of the underlying geometric or probabilistic context. The key is that the randomness comes from independent, identically distributed indicators, and the estimator's variance is determined by both the underlying probability and the scaling applied.
Practise this question with written feedback, or hear it in a spoken mock interview.
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